Connected sums of Gorenstein local rings

A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\leftarrow S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the \emph{connected sum} R#_TS is defined to be the local ring obtained by factoring out the diagonal image of $V$ in the fiber product $R{\times }_{T}S$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is R#_TS. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra \Ext^*_{R#_kS}(k,k) is an amalgam of the algebras ${\text{Ext}}_R^{\ast }(k,k)$ and ${\text{Ext}}_S^{\ast }(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring R#_TS almost never is complete intersection.